I had a funny realization this weekend at my second job as an SAT preparation tutor.

I realized that the students I was tutoring knew just as much about math as I did, if not more. I was on the “slow track” for math in school: I did make it to pre-calculus in my senior year, but I have completely blocked it out (are there diagrams involved in calculus? I seem to vaguely remember graphing things). I also never took a math class while I was in college. I had only gotten about as far as they had: junior-year algebra. And yet when they faced a difficult math problem in the homework or on a practice test, they brought it to me and I could usually figure it out without finding the answer in the back of the book.

I couldn’t understand why, if they weren’t able to do these problems, I was. It wasn’t because I had done the problem before–I’ve only taught this course a few times, and there are so many practice problems, I hadn’t even begun to work my way through them all. It wasn’t because I am instinctively good at the SAT style of testing math, the way I am with the SAT style of testing reading and writing. In fact, my own SAT math score was so low, it almost disqualified me from being hired by this tutoring company. When I do figure out how to solve the problems, I can never quite believe I did it. It always surprises me. I wondered what could possibly account for the ability gap between me and my students?

How Math is Tested on the SAT

In order to understand the disparity, you have to understand a few things about how the SAT tests math. If you’ve taken the SAT, you may remember that the easiest questions come first, with each subsequent question getting harder, until you reach the end of the section, where the highest difficulty questions are. Surprisingly, the highest difficulty questions are solved using the same basic math skills that are used to solve the easy questions: no need to know trigonometry, calculus, or game theory to answer these questions. The only thing that makes them “harder” is that there are more steps–more chances to trip up, to make an error, to get confused.

Another surprise is that there are often little tricks and ‘hacks’ built into each problem. I tell my students that if they’re looking at a problem and thinking, “Oh man, this is gonna take forever to solve,” they are probably missing something. See, the SAT rewards those who think flexibly about numbers. If the test designers really wanted to evaluate math skills, they wouldn’t let students bring calculators. Especially when you get into more difficult questions, SAT math is all about strategy and how you think about math. If you can figure out what they’re asking for, and mentally create a mathematical map to find it, you can solve the problem. How you approach the problem is the key.

What Are You Trying to Prove?

I have the luxury of approaching the problems with an open, curious mind. I even look forward to the challenge of solving an unfamiliar high-difficulty problem. I know that if I can’t figure out, I’ll just look it up in the back of the book and walk the student through the book’s explanation. All that my students are able to think about is the effect their SAT score will have on their college admissions, or how disappointed their parents will be if they get a low score. They believe that if they can’t figure it out, the implicit judgement will follow them around for the rest of their careers.

It became very clear to me that other people’s expectations of us affect our performance, for better or for worse. When I look at a difficult test question, I generally think, “Oh no…this one looks really tough. Maybe I should just flip to the explanation in the back now.” But then I take a deep breath and remember: I am the teacher. I am supposed to be smart enough and capable enough to figure this out; that’s why this company decided to hire me. So even if I feel confused or intimidated, that vote of confidence gives me the motivation to put pencil to paper and muddle through. It gives me the courage to try, and keep on trying until I get the right answer (or at least several wrong ones).

My students, on the other hand, are approaching the problem from a very different perspective. First of all, while I know that I am there to help, my students know that they are there to be helped. This may encourage them to view themselves as, well, helpless. Secondly, the process of being tested puts students in the uncomfortable spot of having to prove their own intelligence. When they get to the high-difficulty questions, the test is whispering to them, “Here’s where we separate the smart kids from the dumb ones. So go ahead, see if you can solve it. Which pile will you end up in?”

Brain Freeze

In his book How Children Fail, John Holt talks about the tension we experience when we are trying to finish something without making any mistakes. He realizes that some of his students are making mistakes on purpose to break the tension.

“Worrying about mistakes is as bad as–no, worse–than worrying about mistakes they have made. Thus, when you tell a child that he has done a problem wrong, you often hear a sigh of relief. He says, “I knew it would be wrong.” He would rather be wrong, and know it, than not know whether he was wrong or not…When the paper was turned in, the tension was ended. Their fate was in the lap of the gods. They might still worry about flunking the [test], but it was a fatalistic kind of worry, it didn’t contain the agonizing element of choice, there was nothing more they could do about it. Worrying about whether you did the right thing, while painful enough, is less painful than worrying about the right thing to do.”

I think this same relief of tension manifests in SAT takers when they leave an answer blank. Whenever the student brings their question to me, the rest of the problems may be marked up, with their work written out, but the difficult problem is always spotless. I admit I haven’t been doing this very long, but I have never seen a student get stuck in the middle of one of these math problems. When I have faced really difficult problems in my student years, it always felt like some kind of mental paralysis: I’d try frantically to figure out what to do, but all I could think was, “I don’t know. I just don’t know!” I couldn’t figure out where I was going, how to get there, or even how to begin.

Solving a difficult SAT math question hinges on approaching it properly: you have to look at what the problem says, what it asks for. You have to think about how to use the information given to get from point A to point B. You have to clear your mind and let the numbers and figures speak to you. If you can’t get to that open, curious, relaxed-yet-alert state of mind, you won’t be able to figure out how to approach the problem, and you’ll be sunk. You’ll hand me your paper, saying helplessly, “I didn’t know where to start.”

Thawing Out

I think the only thing that really helped me out of my math anxiety was knowing that I’m no longer judged by my math skills or lack thereof. I’ve relaxed enough to be able to treat them as intriguing challenges, fun ways to stretch my mind. I hate that I can’t give my students the same permission not to worry about it so much. Also, since I haven’t prepared the problem ahead of time, I can’t really “lead” the student through it. I kind of turn the problem over and over in my head, and then once I’ve got it, I hand it to the student and say, “There.” I don’t think that’s really the eye-opening learning experience they need.

Have you suffered from math anxiety? Have you ever helped any one through it? What are your strategies for helping students move from fear to curiosity to delight?

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Kids are under enormous pressure to do well on tests today. When I was in high school there were socially acceptable options to college if you didn’t test well in school. Now it’s college or failure–at least that’s the inference. The real world is quite different, so it’s nearly criminal the way kids are under that kind of pressure.

Math always presents a special challenge on tests because of it’s absolute nature; your answer is either right or wrong, there’s no partial credit when a machine is doing the grading. In other subjects, you can make a convincing point being only partially right, and you do this by emphasizing what you DO know. That means you’re not dumb–and few people are, they’re just made to feel that way.

Testing, math in particular, brings to mind something J.D. Roth at getrichslowly.com often says, “the perfect is the enemy of the good”. If only we could truly grasp that concept in our increasingly technocratic world.

Oh, yes, I wish you could get partial credit on the SAT. They are really very tricky about it though: the test makers will often put “trap” answers in the multiple choice for high-difficulty problems. For example, say the steps in a given problem require you to solve for x, then use that value to solve for y. The value you find for x might be 5, but when you plug it in and solve for y, the correct answer is 9. Sometimes students forget that last step and just bubble in the answer choice for ‘5’ which is invariably among the choices. Sneaky little buggers…they punish you for trying!

Check out this story by Daniel Greenberg of the Sudbury Valley School in Framingham, Massachusetts:

Shows it’s more about motivation than almost anything else. If one is motivated to learn high school to college level math skills, then one will do so. Whether it’s to pass the SAT or ACT, or to ace a college math placement exam, such as a Compass exam. Furthermore, one can do so, on one’s own, without any classroom teacher, especially today. Good books are available online to download for free that go into every step of every problem, so that the student at any age can go through the steps and understand how to solve the problem. Online math tutors are available free of charge, too. Whole free courses on specific maths are available from all over the world. And top professors courses on topics like “Calculus Made Clear” are available, either free (like MIT’s Open Course Ware) or to purchase from online sources like The Teaching Company. These courses are sold at retail, but are also offered at discounted prices throughout the year, or good used copies are often available from Amazon.com or other similar sources. Whatever the price, these are much cheaper than college courses at per credit prices. For lower levels of math try The Standard Deviants many DVDs, often available from libraries. And most library databases offer free individual courses of study for most high school or pre-high school math subjects, as well as all other required subjects. Look for those from The Learning Company.

So, whenever and if ever someone feels motivated to learn these skills, today’s online and computer-based options instantly open doors to most levels of mathematical learning, at any age.

Thanks so much for letting us know about all the resources out there for learning math! I think the best resource of all is the first one you mention: motivation to learn. I think that is one of the biggest problems my students face: they pretty much only look at math as something to “get through” so they can get a certain score. We are really divorced from the reasons our ancestors first invented and studied math in the first place.

When it was time for my daughter to face the same issues that Daniel Greenberg’s students faced, I remembered his story and presented her with the same opportunity. She took it and spent about twenty weeks completing all her high school level math, up to and including basic calculus and trig. She did it on her own, with all the resources I cited in my previous comment. I was available to her, but I felt that it was better that she struggled through the problems, so that, in the end, she knew she did it herself.

She shed a few tears, screamed in frustration periodically, but she completed it all, able to figure out the problems herself. That meant that when she took the Compass math placement test at the first college she chose to attend, she knew she could do it herself. The particular Compass test they offered at that college worked beautifully. They started with the most difficult problems first. If the student could solve the most difficult ones, then the test was over, passed, fini. In this college students were seated in separate study carrels in a quiet, private space, could schedule the test at their convenience, anytime before the school year began. Those who couldn’t solve the more difficult problems worked backwards until they had worked to their level of understanding. My daughter was done in about ten minutes, told that she would not have to take any math during college. Yay! She was ready to celebrate on that issue alone. I mean, why should a fine arts major need to take college level math in the first place? Since 75% of college admissions do end up taking remedial math classes, this made her feel very good about her twenty weeks of self-directed high school math. Most of the other students she met had placed in remedial math.

By contrast, another college that had offered her a full-tuition scholarship, started with basic math, with students working their way up to higher math on their placement test. She hated that test, forced to take it in a crowded theater-style room, with other students looking over her shoulders and bumping elbows with her. She did not score well on that test, nor in that situation. They recommended remedial math. She complained about the testing situation to them. They said she had math anxiety, definitely needed the remedial math. She promptly turned down that scholarship.

The college she chose to attend was a prestigious state university that also offered her a full-tuition scholarship. But after two years on the Dean’s List, excelling at everything from environmental geology to microbiology, as well as her many studio art courses, she decided to look for something more appropriate for a self-directed learner. She found exactly what she wanted in a totally self-directed online degree program. They did require one math unit from her. She chose “sacred geometry” and loved it, setting up her own study course that revolved around the many aspects of math that really did relate to her creative work. And when they wanted her to earn some credits studying psychology and sociology, she asked if she could do CLEP exams for those subjects. They said yes and she studied a few weeks, took the exams at a college nearby, and completed those credits. After all, if she could do it with math, she knew she could do it with any subject. Motivation is truly the key.

Once upon a time I did really well on the SAT with only middle-school math. I couldn’t understand how high-schoolers, who’d had four years more math, could get worse scores than I had. Even though they claim it’s no longer an IQ test, it’s still more about problem-solving than about content. I can do the problems, and my students know the material, but I have trouble explaining to them how they should know how to solve it.

Yeah, it’s a very interesting story about how the SAT originated as an “IQ test.” Then eventually Stanley Kaplan figured out you could study for it, so it couldn’t really be said to test innate intelligence. But the high-difficulty problems are tricky, because it’s not such a straight-forward, “when you see this, do that” type of process. So it can be hard to teach students what to do. What have you tried?